The purpose of this study will be to evaluate the effectiveness and safety of human growth hormone(hGH) administration in children and adolescents who have had heart transplants and are failing to grow normally. hGH is currently approved for use by the FDA for children with growth hormone deficiency (GHD) and we will be giving it only to those heart transplant patients who are exhibiting growth failure. All enrolled patients will receive the study treatment of hGH at the same dose based on body weight. Each patient will serve as his own control based on a pre-treatment observation of 12-18 months. The study will involve the daily subcutaneous administration of hGH to the patient and the careful monitoring of both growth and possible side effects. The monitoring aspect will be included in the routine every-three-month follow-up that all pediatric transplant patients undergo. These visits include extensive cardiologic testing (including a cardiac biopsy), blood tests, x-rays and a full physical exam.